A231168 Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6*x + 1, 6*y - 1, 6*z -1 are all prime.

0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 3, 2, 5, 1, 4, 3, 2, 3, 1, 1, 4, 2, 5, 3, 3, 4, 4, 8, 2, 3, 8, 2, 4, 3, 4, 8, 7, 2, 2, 8, 3, 8, 6, 1, 6, 8, 4, 1, 9, 2, 4, 10, 6, 1, 7, 11, 7, 10, 2, 6, 9, 3, 6, 3, 6, 6, 6, 8, 4, 8, 4, 4, 9, 2, 11, 4, 9, 6, 1, 4, 5, 5, 10, 7, 5, 6, 6, 7, 5, 8, 17, 8, 5, 2, 7, 8, 11, 10, 6, 4

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 5.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, A conjecture involving squares and primes, a message to Number Theory List, Nov. 5, 2013.

Example

a(19) = 1 since 19 = 13 + 5 + 1 with 13^2 + 5^2 + 1^2 + 1 = 14^2, and 6*13 + 1 = 79, 6*5 - 1 = 29, 6*1 - 1 = 5 are all prime.
a(444) = 1 since 444 = 76 + 28 + 340 with 76^2 + 28^2 + 340^2 + 340 = 350^2, and 6*76 + 1 = 457, 6*28 - 1 = 167, 6*340 - 1 = 2039 are all prime.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Sum[If[PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6(n-i-j)-1]&&SQ[i^2+j^2+(n-i-j)^2+(n-i-j)], 1, 0], {i, 1, n-2}, {j, 1, n-1-i}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000290, A227877, A230121, A230596, A230747.

Sequence in context: A118144 A136691 A361199 * A276830 A064741 A281659

Adjacent sequences: A231165 A231166 A231167 * A231169 A231170 A231171

Keyword

nonn

Author

Zhi-Wei Sun, Nov 04 2013

Status

approved